Unsupervised Learning Methods

Transcription

1 Structural Health Monitoring Using Statistical Pattern Recognition Unsupervised Learning Methods Keith Worden and Graeme Manson Presented by Keith Worden The Structural Health Monitoring Process 1. Operational evaluation 2. Data acquisition & networking 3. Feature selection & extraction 4. Probabilistic decision making Outlier Analysis Statistical Process Control Projection Methods :2

2 Outline Novelty detection. Outlier analysis. Gaussian distribution. Novelty detection on an aircraft structure. Novelty detection is not damage detection. Statistical Process Control. Visualisation and projection methods. 3 Pattern Recognition Pattern Processing Probability Theory Novelty Detection Classification Regression Neural Networks 4

3 Novelty Detection Novelty detection is used to address Level One in Rytter s damage identification hierarchy. Level One: Damage Detection Level Two: Damage Location Level Three: Damage Assessment The method gives a qualitative indication that damage might be present in the structure. The method gives information about the probable position of the damage. The method gives an estimate of the extent and/or type of damage. Level Four: Prediction The method offers information about the safety of the structure (e.g. residual life). 5 Novelty Detection A novelty detection algorithm is required to simply indicate if the acquired data comes from the normal operating condition or not. Two-class problem. This can be done in an unsupervised manner in that labels do not need to be attached to the normal condition data which will be assumed to train the novelty detector. There are many novelty detection techniques, e.g. outlier analysis, kernel density estimation and auto-associative neural networks. All techniques fit a probability distribution to the normal condition data then assess the probability of the test data having been generated by the same mechanism. 6

4 Novelty Detection Sensor (Training Data from Unfaulted Structure) Sensor (Testing Data from Unknown Condition) Pre-Processing Feature Extraction Pre-Processing Feature Extraction Feature Component 2 Test Data Post-Processing Post-Processing Feature Component 1 Pattern Recognition (Novelty Detection) Decision (Undamaged) (Damaged) Testing Data Point Threshold 7 Novelty Measure Outlier Analysis Discordant outlier is data point which is surprisingly different from rest of data and therefore believed to be generated by an alternate mechanism to the other data. In damage detection, the data point is different to the data acquired from the normal operating condition of the structure and this is assumed to be due to the presence of damage. Outlier detection can be conducted for both univariate and multivariate data sets and is based upon deviation from a Gaussian distribution. 8

5 Outlier Analysis Gaussian Distribution Also known as the normal distribution. The most widely used model for the distribution of continuous variables. In the case of a single variable x, the Gaussian distribution can be written: px ( ) exp ( x) where is the mean and is the standard deviation. Gaussian distribution for Gaussian distribution for 0 and and 1 9 Outlier Analysis Gaussian Distribution For a D-dimensional vector x, the multivariate Gaussian distribution can be written: 1 1 T 1 px ( ) exp ( ) D/2 1/2 1 x μ 1x μ px T 1 2 p ( x) 2 exp ( ) ( ) D/2 1/2 x x (2 ) 2 where μ is a D -dimensional mean vector and is a D D where is a mean vector and is a covariance matrix. covariance matrix, and denotes the determinant of. Contours of constant probability density for 2- dimensional Gaussian distribution (a) general covariance matrix, (b) diagonal covariance matrix and (c) covariance matrix proportional to identity matrix. 10

6 Outlier Analysis Univariate test for discordancy: x zz where x x is is the the candidate outlier, outlier is and the mean and are the of mean the normal and standard condition deviation data and of is the it's training standard sample. deviation Multivariate test for discordancy (Mahalanobis distance): T 1 D2 x μt 1 x μ D ( x ) ( x ) where where x x is is the the candidate candidate outlier, outlier is the and mean μ and vector are the of mean the normal vector condition and covariance data and matrix is it's covariance of the training matrix sample. 11 Outlier Analysis The procedure for outlier analysis is as follows: Acquire training data (univariate or multivariate) from normal operating state of machine or structure. Pre-process data then extract features. Calculate Gaussian statistics for normal condition features. and in the univariate case and and in the multivariate case Acquire test data, pre-process and extract same features as previously. x in the univariate case and in the multivariate case x 2 ( z or D ) Calculate discordancy value and compare with statistically calculated threshold value. If greater than threshold, damage is inferred. 12

7 Outlier Analysis The threshold values are calculated using a Monte-Carlo approach based on extreme value statistics. Threshold is dependent upon the number of dimensions of the feature and the number of observations in the normal condition data set. The threshold can be calculated for a number of critical values (e.g. 1% and 0.1% tests of discordancy). 13 Novelty Detection on an Aircraft Structure Test Structure Experimental validation of SHM methodology based on novelty detection. Philosophy of programme was to develop methods which are robust enough to be successful on real aircraft structures. 14

8 Novelty Detection on an Aircraft Structure Data Capture Not possible to damage aircraft. Damage introduced in inspection panel. Base measurements transmissibilities. 4 sensors used in 2 pairs. Wing excited with white Gaussian excitation. 110 unfaulted transmissibilities and 10 from each fault recorded. 15 Novelty Detection on an Aircraft Structure Novelty Detection Two combined features capable of detecting all damage cases whilst correctly classifying unfaulted patterns. Robust to some extent. Outlier analysis results for feature from spectral lines of T12. Outlier analysis results for feature from spectral lines of T34. 16

9 Novelty (not Damage) Detection Novelty Detection has shown much promise in providing low-level damage detection strategies. Problem identifies if a machine or structure has deviated from normal condition. Deviation from normal condition does not always signify damage. Effect of using inappropriate normal condition set is potentially disastrous. Data from undamaged structure could be flagged as damage and data from damaged structure could go undetected. How do we make the damage detection process robust? 17 Novelty (not Damage) Detection Novelty detector trained on N1 data should easily classify T1 as novel. Feature Component 2 Feature Component 1 18

10 Novelty (not Damage) Detection Novelty detector trained on N1 data N3, N4 and N5 likely to be classed as novel. Feature Component 2 Feature Component 1 19 Novelty (not Damage) Detection Feature Component 2 Feature Component 1 Three main options for dealing with problem. Train novelty detectors using data from all normal conditions. Train novelty detector for each normal condition. Test against relevant detector. Transform feature to remove sensitivity to normal variation whilst retaining sensitivity to damage. 20

11 Statistical Process Control (SPC) SPC originated in chemical and process industries as a means of determining when a process leaves normal operation so that a remedial control action can be taken. Features are samples from time-series, in simplest case one has univariate measurements x(t). Simplest approach is to use outlier analysis with the normal condition set assembled from in-control samples. The samples are plotted on a chart the X-chart - with upper and lower control limits (UCL and LCL) set at k around the mean for in-control data. Note that setting k = 3 gives the 99.7% confidence limits for an assumed Gaussian in-control set. 21 SPC II This approach is not robust against process noise. Sensitivity is enhanced by averaging over blocks of N samples and plotting these on the chart the X-bar chart. UCL and LCL become k / N above and below mean of data and sensitivity is enhanced. As an example consider bookshelf data. In control undamaged state Nonlinear impacts damaged state 22

12 Other Control Charts S-chart monitors variance. Cumulative Sum or CUSUM chart. Exponentially Weighted Moving Average (EMWA) chart see Montgomery (1996) for details. There are multivariate versions of all the charts the multivariate version of the X-chart is essentially the multivariate outlier analysis described earlier. 23 Visualisation - Projection Methods It is very often useful to make a visual assessment of data. The SPC control charts are an excellent example of this; departures from control are immediately visible as points cross the control limits. For multivariate outlier analysis or SPC (MSPC), things are much more complicated as human physiology precludes visualisation of anything higher than 3-dimensional. The solution is to project data from the high-dimensional feature space into a low-dimensional (2 or 3D) space so that a plot can be made. Such projections lose information from the original data, alternatively one can think in terms of them only preserving certain information. 24

13 Principal Component Analysis (PCA) Mainstay of multivariate statistical analysis. PCA allows projection of data from p-dimensions to q- dimensions where q < p. An advantage of PCA is that the low-dimensional variables are uncorrelated. What PCA preserves is power, or alternatively variance. It is a linear transformation. Another linear projection method is Factor Analysis; this preserves (as far as possible) the covariance structure of the data. Factor analysis is a generalisation of PCA. Both algorithms have nonlinear variants. Another nonlinear projection method of interest is Sammon Mapping; this preserves (as far as possible) the metric distance between data points. 25 Example of PCA Actuator Bracing Restraints Test Piece Sensor Typical AE signal observed from Girder. Box Girder under test; under load generates acoustic emissions. 26

14 Example of PCA II Multivariate features are extracted in order to characterise AE bursts. Too many features to visualise, PCA reduces to 2D. 27 Example of PCA III Second principal component Cluster 1 Cluster 2 Cluster First principal component On subsequent analysis it was found that those observations in cluster 1 arose from crack related events, all those in cluster 2 from frictional processes away from the crack and all those in cluster 3 from crack-related events detected at a distance away from the sensor. Visualisation here has lead to an interpretation of the data. 28

15 Example of PCA IV Example of each of the three distinct AE signals. 29 Summary Unsupervised learning is the form of learning used when class labels for data are not known. It is appropriate to damage detection when only examples of the normal condition are known. Novelty detection methods test for deviation from a learned normal condition outlier analysis is one of the most straightforward but assumes a Gaussian normal condition. Novelty detection is not damage detection; one must take care that one is not inferring damage from a benign change. Statistical Process Control (SPC) techniques essentially implement outlier detection for time series data. Visualisation of data can be very informative; dimension reduction techniques like PCA are usually needed. 30

16 References V. Barnett and T. Lewis. Outliers in Statistical Data, 3 rd Edition. John Wiley and Sons, Chichester, G. Manson, K. Worden, K.M. Holford, R. Pullin, A. Martin and D.L. Tunnincliffe (D.L.) Visualisation and dimension reduction of acoustic emission data for damage detection Journal of Intelligent Material Systems and Structures 12 pp M. Markou, and S. Singh. Novelty detection: a review. Part 1: statistical approaches. Signal Processing, 83, pp , 2003a. M. Markou, and S. Singh. Novelty detection: a review. Part 2: neural network based approaches. Signal Processing, 83, pp , 2003b. D.C. Montgomery. Introduction to Statistical Quality Control, John Wiley & Sons, Inc., New York, S. Sharma. Applied Multivariate Techniques. John Wiley and Son, K. Worden. Structural fault detection using a novelty measure. Journal of Sound and Vibration, 201, pp , K. Worden, G. Manson and N.R.J. Fieller. Damage detection using outlier analysis. Journal of Sound and Vibration, 229 pp , K. Worden, G. Manson and D.J. Allman. Experimental validation of structural health monitoring methodology II: novelty detection on an aircraft wing. Journal of Sound and Vibration, 259 pp.344,